We study the contact Floer homology introduced by Merry–Uljarević in , which associates a Floer-type homology theory with a Liouville domain W and a contact Hamiltonian h on its boundary. The main results investigate the behavior of under the perturbations of the input contact Hamiltonian h. In particular, we provide sufficient conditions that guarantee to be invariant under the perturbations. This can be regarded as a contact geometry analog of the continuation and bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer homology in symplectic geometry. As an application, we give an algebraic proof of a rigidity result concerning the positive loops of contactomorphisms for a wide class of contact manifolds.